andrederiverten

Andrederiverten denotes the second derivative of a function, the derivative of its first derivative. If a function f is differentiable on an interval and its derivative f' is differentiable there, the second derivative exists and is written as f''(x) = d^2f/dx^2. It measures how quickly the slope of f is changing.

Example: for f(x) = x^3, f'(x) = 3x^2 and f''(x) = 6x.

Interpretation: The sign of f'' indicates concavity: f''(x) > 0 means f is concave up at x; f''(x) < 0 means concave down. Points where f''(x) = 0 or undefined can be inflection points where concavity changes, though not always.

Relationship to curvature: for a graph y = f(x) with a continuous second derivative, the curvature κ at x is κ = |f''(x)| / (1 + f'(x)^2)^(3/2).

Notation and higher order derivatives: f'' can also be denoted f^(2)(x). Higher derivatives include f''' and beyond. In physics, the second derivative with respect to time represents acceleration; in optimization, the second derivative test uses f'' to classify critical points.

See also: derivative, higher-order derivative, Taylor series, inflection point, curvature.

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